Generalizations of string theory

(talk at Generalized Geometry & T-dualities, 5/11/16)

Outline

T-theory

1. Current algebra: strings → double field theories
2. New spaces: field strengths as gauge fields
3. Chiral strings: boundary condition → finite spectrum
   1. Effective actions: new for (massive) gravity
   2. Particle amplitudes: scattering eqs. & KLT

(not your father's) F-theory (nor mother's M-theory)

1. Current algebra: branes → exceptional gravity
2. 0th-quantized ghosts

T-theory

Current algebra: strings → double field theories 👯

T-duality (worldsheet, no background) (WS '83)
Fields d=1 dimensionally reduced (Buscher '87)
d>1: O(d,d)/O(d)² scalars (Duff '89, Tseytlin '90)
"Double field theory", bosonic & heterotic (WS '93):

• O(D,26)/O(D−1,1)O(25,1); D = 10,26;
  for gauge fields g,B,A (spont. broken O(D,26))
• "strong constraint" (sectioning) ↔ (gauge inv.)²
• "C-bracket" & "D-bracket" (nonabelian for het.)
• ∇_A, T_ABC, R_AB^cd, field equations, action
• gauge fixing: LR factorization of Feynman rules

Our recent work (see MH talk for some details)

Spin S_ab & dual Σ_ab (WS '11, Poláček & WS '13)

• ⊕ current super algebra D_α,P_a,Ω^α (WS '85)
• curvature as torsion
• Lorentz connection couples to string

Type II & RR (Hatsuda, Kamimura, WS '14-'15)

• RR gauge fields in central currents Υ_αα',Ϝ^αα'
• Type II algebra from string action

3D Type II off-shell background (Poláček & WS '14)

• prepotential in vielbein without derivatives on it
• O(3,3)/O(2,1)² (= O(2,2))
• O(3,1) M-theory & O(3,2) F-theory (see later)

New spaces: field strengths as gauge fields (WS '11)

T-theory methods generalize to particle field theory, e.g., gravity

• S_ab: 1st-quantization of spin
• spin coordinates ⇒ curvature as torsion
  (tangent space as subspace of enlarged manifold)

But also Yang-Mills

• Σ_ab: field strength F_ab = gauge connection for Σ
• first-order action = PPΣ Chern-Simons

Chiral strings: b.c. ⇒ finite spectrum (see BZ talk)

Effective actions (Hohm, Zwiebach, WS '13)

• chiral (non-conformal) gauge: X(z) (no z̄)
• chiral boundary conditions: N+N̄=0 (not −)
• Virasoro algebra ⇒ field equations
• complete @ 6 derivatives (α'²) ⇒ ...+R³
• α' modifications to brackets
• unconstrained fields (not O(26,26))
• cubic action (except for dilaton)
• like 2D CP(n), nonlinear σ → linear @ O(α'):
  Lagrange multiplier → massive, propagating
  (understood recently: see below)

Particle amplitudes for all dimensions

"Scattering equations" (Cachazo, He, Yuan '13)

From string-like theories (Mason & Skinner '13)

Derivation from usual string theories (WS '15)

• chiral boundary conditions (HSZ)
• "almost" chiral gauge (HSZ): z̄ as IR regulator
• usual string vertex operators

KLT factorization (Huang, Yuan, WS '16)

• chiral boundary conditions, but conformal gauge
• ordinary open strings, except η_ab → − η_ab for z̄
• sign change cancels 0's against usual poles
• fixes earlier bosonic failure: (few) massive modes

F-theory (& M-theory) (Linch & WS '15)

Before branes: more-than-DFT for exceptional SG
(Coimbra, Strickland-Constable, Waldram '11;
Berman, Cederwall, Kleinschmidt, Thompson '12)

Current algebra on fundamental branes

• brane background: exceptional (super)gravity
• X: selfdual gauge fields on worldvolume
• worldvolume indices are spacetime indices
• worldvol. metric constrained ⇒ nonpropagating
• sectioning for worldvolume (Gauss's law)
• sectioning = 0-modes of dimensional reduction
• solve Virasoro → M-theory; Gauss → T-theory

0th-quantized ghosts (WS '16)

• 〈X X〉 = -ln z² for higher-d z (branes)?
• ghosts for every order of quantization:
  2. usual ghost fields for g(X),B(X), etc.
     (2nd-quantization)
  1. worldvolume ghost "fields" c,b for X(z)
     (1st-quantization)
  0. ghost coordinates for z
     (0th-quantization)

Appendix: Some technical details

T-theory current algebra with Ramond-Ramond

Currents: (S_ab,D_α,P_a,Ω^α,Σ_ab)_±;Υ_αα',Ϝ^αα'

{D_±,D_±} = P_±δ, {D₊,D₋} = Υδ

[D_±,P_±] = Ω_±δ, [D_±,Ϝ] = Ω_∓δ

[S_±,not Σ] ~ not Σ, [S_±,Σ_±] = Σ_±δ + δ'

[P_±,P_±] = Σ_±δ + δ', [Υ,Ϝ] = (Σ₊+Σ₋)δ + δ'

{D_±,Ω_±} = Σ_±δ + δ'

1st-order Yang-Mills as Chern-Simons

With torsion [d_A,d_B} = T_AB^Cd_C, the CS form is

X_ABC = ½A_[Ad_BA_C) −¼A_[AT_BC)^DA_D +⅓iA_[AA_BA_C)

In particular, for

[p_a,p_b] = σ_ab ⇒ T_a,b^cd = ½δ_[a^cδ_b]^d

we choose

L = ½a^a,b,cdX_a,b,cd , a^a,b,cd = η^a[cη^d]b

Then, labeling A_p→A, A_σ→F,

L ~ AσA + FpA − FF + FAA

gives the usual 1st-order action after σ→0.

Scattering equation δ from almost-chiral gauge

Change in boundary condition,

Δ = − ln z − ln z̄ → − ln z + ln z̄

followed by change in gauge (with β→∞),

→ − ln z + ln (z̄ − βz) → −

1

β

z̄

z

modifies z̄ half of Koba-Nielsen integral

∫dz̄_i exp

1

2β

∑k_i∙k_j

z̄ij

zij

~Π_i δ(∑_j

ki∙kj

zij

)

KLT for chiral string (toy example, m=0)

Normal → chiral [sin(πt) = −π/Γ(−t)Γ(1+t)]:

−

Γ(−s)Γ(−t)

Γ(u)

1

Γ(−t)Γ(1+t)

Γ(1−t)Γ(−u)

Γ(1+s)

=

1

s

Γ(−s)Γ(−t)Γ(−u)

Γ(s)Γ(t)Γ(u)

→ −

Γ(−s)Γ(−t)

Γ(u)

1

Γ(−t)Γ(1+t)

Γ(1+t)Γ(+u)

Γ(1-s)

=

1

s

F-theory dimensional reduction & sectioning

σ-Virasoro	S ≡ PP
dim. reduction	S̊ ≡ pP	U ≡ ∂P
sectioning	S̥ ≡ pp	U̥ ≡ ∂p	V ≡ ∂∂

P = bosonic current (selfdual field strength);
p = 0-mode, ∂ for worldvolume

Downward: P→p (otherwise same)

Dimensional reduction kills X's (U = Gauss)

Section conditions kill 0-modes (background) & σ's

(S̊,S̥) = 0 ⇒ M-theory; (U,U̥,V) = 0 ⇒ T-theory

A little F-theory current algebra

d worldvolume coordinates: τ,σ_M

Spacetime coordinates:Θ^µ,X^m,Y^m',Ỹ_Mm'
(32 Θ's, <10 worldvolume scalars Y, their duals Ỹ)

Currents: (S,)D_µ,P_m,Υ_m',Ϝ^Mm',Ω^Mµ(,Σ)

[▷_M , ▷_N} = f_MN^P▷_Pδ + 2iη_MNR∂^Rδ

⇒ f_[MN|^Qf_Q|P)^R = 0 = f_M(N|^Qη_Q|P]R

S^R = η^MNR▷_N▷_M

f's ~ super YM, but doubled spinor indices

Types of F-theories (N=1 D-dim. SG bosons in G/H)

D	d	H	G	X	σ
1	3	GL(1)	GL(2)	2+1	2
2	4	GL(2)	SL(3)SL(2)	(3,2)	(3',1)
3	6	Sp(4)	SL(5)	10	5'
4	11	Sp(4,C)	SO(5,5)	16	10
5	28	USp(4,4)	E6(6)	27	27'
6	134	SU*(8)	E7(7)	56	133

G = "E_D+1",
H = covering of SO(D−1,1) with double argument,
H' = SO(10−D)

Y = vector of H', Θ = spinor of H⊗H' (32)